Numerical Solution to Generalized Burgers'-Fisher Equation Using Exp-Function Method Hybridized with Heuristic Computation

Abstract

In this paper, a new heuristic scheme for the approximate solution of the generalized Burgers'-Fisher equation is proposed. The scheme is based on the hybridization of Exp-function method with nature inspired algorithm. The given nonlinear partial differential equation (NPDE) through substitution is converted into a nonlinear ordinary differential equation (NODE). The travelling wave solution is approximated by the Exp-function method with unknown parameters. The unknown parameters are estimated by transforming the NODE into an equivalent global error minimization problem by using a fitness function. The popular genetic algorithm (GA) is used to solve the minimization problem, and to achieve the unknown parameters. The proposed scheme is successfully implemented to solve the generalized Burgers'-Fisher equation. The comparison of numerical results with the exact solutions, and the solutions obtained using some traditional methods, including adomian decomposition method (ADM), homotopy perturbation method (HPM), and optimal homotopy asymptotic method (OHAM), show that the suggested scheme is fairly accurate and viable for solving such problems.

Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

“…Malik et al [29] developed a heuristic scheme for the numerical solution of the GBF equation based on the hybridization of Exp-function method with nature inspired algorithm. The problem was converted into a nonlinear ordinary differential equation (ODE) by substitution.…”

“…After simplification dividing both sides by ℎ , we obtain the following expressions given in (29) and (30) for GBF and GBH equations, respectively:…”

“…The absolute errors calculated at different time levels by proposed HBSCM are compared with the existing methods Exp-function method (EFM) [29], improved algorithm (IA) [37], ADM [40], OHAM [69], and Nonlinear Analytic Technique (NAT) [70] in Table 16. It is concluded that the proposed method is more accurate than these methods.…”

“…It is concluded that the proposed method is more accurate than these methods. Maximum error at t = 0.5 The absolute errors are calculated by proposed HBSCM and compared with the existing methods: computational meshless method (CMM) [27], EFM [29], IA [37], ADM [40], and NAT [70] at two different time levels which are tabulated in Table 17. A remarkable difference in accuracy can be noted from this table.…”

Numerical solution to nonlinear biochemical reaction model using hybrid polynomial basis differential evolution technique

“…In recent years many authors have used nature and biologically inspired based computation techniques as an alternative for solving nonlinear ODEs arising in diverse fields of engineering and science [6][7][8][9][10][11][12][13][14][15][16]. Very recently Malik et al [6,7] employed heuristic technique based on hybrid genetic algorithm for numerically solving the nonlinear singular boundary value problems in physiology and the Bratu problem.…”